Do I get all these vectors into an augumented matrix?
like:
[ -1 1 47 | 7 ]
[ 2 5 -8 | 4 ]
[ -9 1 -7 | 1 ]
and then row reduce completely? I got very awkward numbers and just wanted to make sure I'm using the correct method.

I take it that "relative to these three vectors" means write it as a linear combination of those vectors. What you want are three numbers, a, b, c such that
a<-1, 2, 9>+ b<1, 5, 1>+ c<47, -8, -7>= <7, 4, 1>. Looking at x, y, and z components separately that means -a+ b+ 47c= 7, 2a+ 5b- 8c= 4, 9a+ b- 7c= 1 which gives exactly the augmented matrix you have.